Metastable confinement of particle systems near linearly stable McKean-Vlasov equilibria
Abstract
We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on ℝd with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time T is bounded by C(1+T)e−cN, where N is the number of particles and C,c>0 are independent of N and T. Consequently, confinement holds up to times eγN, for every 0<γ<c, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.
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BibTeX
@misc{chanashing2026metastable,
title = {{Metastable confinement of particle systems near linearly stable McKean-Vlasov equilibria}},
author = {Samuel Chan-ashing and Quentin Cormier},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13655},
eprint = {2610.13655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13655},
}
RIS
TY - UNPB
TI - Metastable confinement of particle systems near linearly stable McKean-Vlasov equilibria
AU - Samuel Chan-ashing
AU - Quentin Cormier
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13655
UR - https://arxiv.org/abs/2610.13655
AB - We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on ℝ^d with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time T is bounded by C(1+T)e^−cN, where N is the number of particles and C,c>0 are independent of N and T. Consequently, confinement holds up to times e^γN, for every 0<γ<c, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.
ER -